Stephane Redon, Nico Galoppo, Ming C....

transcript

Adaptive Dynamics of Articulated Bodies

Adaptive Dynamics of Articulated BodiesAdaptive Dynamics of Articulated Bodies

Stephane Redon, Nico Galoppo, Ming C. LinUniversity of North Carolina at Chapel Hill

Motivation

• Articulated bodies in Computer Graphics– Humans, hair, animals

– Trees, forests, grass

– Deformable bodies

– Molecular graphics

– …

Motivation

• Forward dynamics

• Optimal solutions are linear

• Optimal forward dynamics methods are too slowfor numerous or complex articulated bodies

Contributions

• Forward dynamics

• Adaptive forward dynamics– Specify the number of degrees of freedom

– Only this number of degrees of freedom is simulated

– The most relevant degrees of freedom are automatically found

Contributions

• Hybrid bodies– Articulated-body representation

– To reduce the number of degrees of freedom

• Adaptive joint selection– Customizable motion metrics

– To determine the most relevant degrees of freedom

• Adaptive update mechanisms

Outline

• Related work

• Hybrid bodies

• Adaptive joint selection

• Results

Related workForward dynamics of articulated bodies

• View-dependent dynamics

• Articulated-body motion simplification

– Faure 1999

– Redon and Lin 2005: Adaptive quasi-statics

• Optimal algorithms

• Parallel algorithms

• Human motion

• Plant motion

• Hair modeling

This paper: adaptive simplification using customizable motion error metrics

Outline

• Related work

• Hybrid bodies

• Results

Articulated-bodyDefinitionArticulated-bodyDefinition

• An articulated-body is a rigid-body system with one or more handles;

• A handle is a specified location within an articulated body to which external forces may be applied and which responds with an observable acceleration.

Articulated-bodies Featherstone’s DCA

• Recursive definition

An articulated body is recursively defined as a pair of articulated bodies connected by a joint

• Recursive definition

Articulated bodies Featherstone’s DCA

Rigid bodies

The complete articulated body

Pairs of rigid bodies

The assembly tree of an articulated body

Articulated-body equations of motion:

Articulated-body Dynamics Featherstone’s DCA

BodyAccelerations

Inverse inertias and cross-inertias

AppliedForces

Biasaccelerations

Articulated-body Dynamics Featherstone’s DCA

Articulated-body equationsthe effect of a force applied to body

2, on the acceleration of body 1

The bias acceleration is the acceleration of body 1 when no forces are applied

Featherstone’s DCATwo main passes

1. The main pass: Compute the articulated-body coefficients ( )

Inverse inertias

Bias accelerations

Leaf-node coefficients

Acceleration-independent external force applied to the rigid body

Featherstone’s DCATwo main passes

2. The back-substitution pass: the kinematic constraint forces are propagated down the tree to compute all the joint accelerations ( ).

Joint acceleration

Kinematic constraint forces

Hybrid bodiesDefinitions

• Active region

• Goal: to simply the dynamics

• Means: select a subset of joints to simulate (the complement set of nodes are rigidified)

rigid node

hybrid node

The active region contains the mobile joints

Hybrid body – an articulated body whose set of active joints is a sub-tree of the assembly tree, with an identical root.

Hybrid bodiesHybrid-body coefficients

• Hybrid nodes use the articulated-body equations

• A rigidified node behaves like a rigid body

Rigidify joint

Articulated-body coefficients

Rigidified-body coefficients

Hybrid bodiesHybrid-body simulation

• Same steps as articulated-body simulation

• Computations restricted to a sub-tree in the back-substitution pass (and consequently in updates of the velocities and positions)

Outline

• Related work

• Hybrid bodies

• Results

Adaptive joint selectionAdaptive joint selection

To predict which joints should be activated so as to best approximate the motion of the articulated body, without computing the accelerations of all the joints in the articulated-body.

Adaptive joint selectionMotion metrics

• Acceleration metric

• Velocity metric

Symmetric, PSD

usually identity matrix

Adaptive joint selection Motion metrics

• TheoremThe acceleration metric value of an articulated body can be computed before computing its joint accelerations

Computed in a bottom-up fashion just like the computation of articulated-body coefficients

Adaptive joint selection Acceleration update

Example:

=6 =-3 =2 =-1 =1=-6=3

Adaptive joint selection Acceleration simplification

Compute the acceleration metric value of the root

= 96 -3

Compute the joint acceleration of the root

= 6= 81

Compute the acceleration metric values of the two children

Select the node with the highest acceleration metric value

= 6= 81

Compute its joint acceleration

-6= 81 = 6

= 9 = 36

Compute the acceleration metric values of its two children

-6 = 6= 81

= 9 = 36

-6 = 6= 81

Select the node with the highest acceleration metric value

= 9 = 36

-6 = 6= 81

Compute its joint acceleration

Stop because a user-defined sufficient precision has been reached

Four subassemblies with joint accelerations implicitly set to zero

Outline

• Related work

• Hybrid bodies

• Results

Adaptive update mechanismsHandling two types of coefficients

Limit the update of the coefficients to a subtree

1. Position-dependent coefficientsHierarchical state representation [Redon and Lin 2005]

Adaptive update mechanisms

2. Velocity-dependent coefficients

Linear coefficients tensors:

Outline

• Related work

• Hybrid bodies

• Results

ResultsAdaptive joint selection

Adaptive joint selection example (10x speed-up)

ResultsTime-dependent simplification

One color per sub-assembly

ResultsTime-dependent simplification

One color per sub-assembly

ResultsProgressive simplification of motion

0.25ms

0.02ms

Average cost (ms) per time step

a 300-link pendulum, N- number of active joints

ResultsPrecision / Performance trade-offResultsPrecision / Performance trade-off

# of active joints

# of external forces

ms per interation

ResultsTest application

Stephane Redon, Nico Galoppo, Ming C....

Documents